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Watershed Planning within a Quantitative Scenario Analysis Framework
12:44

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Published on: July 24, 2016

Fractional calculus in hydrologic modeling: A numerical perspective.

David A Benson1, Mark M Meerschaert, Jordan Revielle

  • 1Hydrological Science and Engineering, Colorado School of Mines, Golden, CO 80401, USA.

Advances in Water Resources
|March 26, 2013
PubMed
Summary

Fractional derivatives arise from stable Lévy motion, with fractional integration as their inverse. Numerical methods for these equations reveal the core nature of fractional calculus.

Keywords:
Fractional Brownian motionFractional calculusMobile/immobileSubordination

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Area of Science:

  • Mathematics
  • Physics
  • Stochastic Processes

Background:

  • Classical calculus extensions and their link to natural phenomena.
  • The diffusion equation as a model for Brownian motion.
  • Stable Lévy motion as a fundamental framework for fractional calculus.

Purpose of the Study:

  • To establish fractional derivatives as originating from stable Lévy motion.
  • To define fractional integration as the inverse operator.
  • To explore numerical solutions for fractional calculus problems.

Main Methods:

  • Derivation of fractional integration from governing equations of stable Lévy motion.
  • Development of Eulerian and Lagrangian numerical solutions for fractional partial differential equations.
  • Application of Eulerian methods for stochastic integrals.

Main Results:

  • Fractional derivatives are intrinsically linked to stable Lévy motion.
  • Fractional integration serves as the inverse operator to fractional differentiation.
  • Numerical approximations provide insights into the behavior of fractional calculus.

Conclusions:

  • Fractional calculus is deeply rooted in the principles of stable Lévy motion.
  • Numerical methods are crucial for understanding and applying fractional calculus.
  • The study highlights the fundamental connection between mathematical operators and natural phenomena.